Mathematical Properties – What underpins children’s informal strategies?

This post, which follows on from the last two, forms a series of posts that aims to highlight some of the guidance documents that we’ve included in the new HNP Summary Documents for each year.  If you haven’t read the last two, you can find them here:

Making sense of part-whole strategies for addition and subtraction

Progression in Number Ranges at First Level (First** and First***)

Today’s document, Properties of Addition and Subtraction, has several aims:

  • To raise awareness of the mathematical properties that underpin the informal strategies that pupils use.
  • To highlight some of the main features of the guidance document.
  • To try and bridge a gap between Primary and Secondary.

The document is quite lengthy and it’s not intended that you read it straight through from start to end but use it more as a guide or reference document to support your planning, teaching and understanding of mathematics.

I’ll be honest, when I started teaching, my knowledge of the properties referred to in the document, some of which include:

  • the commutative property,
  • the associative property,
  • the distributive property,

was limited.  I may have had an implicit awareness of them from school, although my schooling was very much of the rules and procedures style and asking why something worked was met with the response that I didn’t need to know why, I just needed to learn it.  I may have covered them very briefly on my teacher training course but again, I’m not particularly aware of them having much prominence there although they are clearly in the main texts that were part of the course: Haylock was one and Frobisher was the author of the other.  I believe these are still the main texts used on courses today… although they may be several editions on!

So, if you read the document and think that some of the content is a bit heavy going… fear not!  I would have also felt that way at one point too.  That being said, I do feel it is fundamentally important that teachers at both primary and secondary have a deep understanding of the properties and how/where they are used.

From experience, although this is purely speculation, a lot of the recurring themes where children get stuck or have misconceptions appear to stem back to a lack of understanding about these mathematical properties.  I believe a lot of this could be mitigated if both pupils and teachers being more aware of the properties.  Algebra would be a good example of this but another good one is pupils frequently trying to solve problems like 52 – 25 but getting an answer of 33.  The document highlights issues like this and connects them back to the properties.Screenshot 2020-08-28 at 09.41.29.png

An additional point to make here is that from a really young age, an implicit awareness of the properties can be fostered as part of play based learning.  Ideas for activities that support a developing awareness of the properties is included within the document – this includes ideas for pupils in Nursery settings but extends to activities for older pupils too.Screenshot 2020-08-28 at 09.42.00

The activities don’t need to be dry, and if teachers really understand the properties, incorporating understanding of them while pupils are exploring different strategies can promote curiosity as to how on earth it’s possible to solve a problem in that way or how someone has solved a seemingly complex problem in a matter of seconds!  Ideas are given about how to support this kind of discussion and how to extend pupils’ understanding of these properties through the development of relational thinking (among other things).Screenshot 2020-08-28 at 09.43.04.png

And there is, hopefully, a clear explanation of what each property is and how it links to the informal strategies that pupils use.  This ties back well do the guidance document on part-whole strategies for addition and subtraction.Screenshot 2020-08-28 at 09.42.42

The final point I want to make is about bridging the gap between primary and secondary.  My feeling is, and this is a generalisation based on my own experiences and certainly not true in every case, that secondary teachers have a good awareness of the mathematical properties but not the informal strategies that pupils use and primary teachers have a better understanding of the informal strategies that pupils use but lack the understanding of the mathematical properties.  If this gap could be bridged, by both parties, then I think primary teachers would be in a better position to help guide the use of pupils’ informal strategies to increasing efficiency and sophistication, through greater awareness of the mathematical properties at play.  If secondary teachers were more aware of the informal strategies that pupils use and the mathematical properties that underpin them, it would become more apparent how these properties are the same properties at play when pupils learn algebra and so these ideas, that pupils often find so hard to grasp, can hopefully be attached to something that pupils are more familiar with and can generalise from.

I’d be interested to know people’s thoughts on this and whether this reflects other people’s experiences or not.

As ever, if anyone spots any mistakes or anything that is not accurate within the documents please do let me know so I can address this.

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